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Maths 1 > Sem 1 Stewart > Flashcards

Sem 1 Stewart Flashcards

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1
Q

function or not?

A

vertical line test

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2
Q

piecewise function

A

defined differently in different parts of the domain

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3
Q

even function or odd function

A

even f(-x)=f(x)
odd f(-x)=-f(x)

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4
Q

increasing functions

A

if f(x1)<f(x2) when x1<x2

vice versa for decreasing

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5
Q

polynomial

A

contains non-negative integers only

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6
Q

power function

A

f(x)=x^a

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7
Q

sec

A

1/cos

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8
Q

cosec

A

1/sin

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9
Q

cot

A

cos/sin = 1/tan

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10
Q

exponential functions

A

f(x)=b^x

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11
Q

log function

A

g(x)=logbx

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12
Q

lnx

A

logex

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13
Q

exponential properties

A

domain real numbers
range (0,infinity)
increasing if b>1, decreasing 0<b<1

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14
Q

secant

A

cuts a curve more than once

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15
Q

left hand limit

A

limit as approaches from the left

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16
Q

right hand limit

A

limit as approaches from the right

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17
Q

limit as x approaches a of f(x) only exists if…

A

Left hand limit=right hand limit

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18
Q

vertical asymptote if

A

lim= -ve or +ve infinity
LHL=-ve or +ve infinity
RHL=-ve or +ve infinity

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19
Q

lim as x approaches a (f(x)+/-g(x))=

A

lim as x approaches a f(x) +/- lim as x approaches a g(x)

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20
Q

lim as x approaches a (cf(x))=

A

c lim as x approaches a f(x)

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21
Q

lim as x approaches a (f(x)g(x))=

A

lim as x approaches a f(x) . lim as x approaches a g(x)

*same for quotient if g(x) does not equal 0**

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22
Q

lim as x approaches a nth root of f(x)=

A

nth root of lim as x approaches a f(x)

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23
Q

if f is a polynomial, a in domain, then

A

lim as x approaches a f(x) = f(a)

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24
Q

squeeze theorem

A

if f(x)</=g(x)</=h(x) x near a

lim as x approaches a f(x)=lim as x approaches a h(x) =L

then
lim as x approaches a g(x)=L

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25
tangent = m = lim as x approaches a of
f(x)-f(a)/x-a
26
if differentiable at a,
continuous at a
27
differentiable on an open interval if
differentiable at every number in interval
28
can fail to be differentiable if:
1. has corner/kink 2. discontinuous 3. has vertical tangent line
29
indeterminant form 0/0
form lim as x approaches a f(x)/g(x) where both f(x) and g(x) approach 0 as x approaches a
30
indeterminant form infinity/infinity
lim as x approaches a f(x)/g(x) where both f(x) and g(x) approach infinity as x approaches a
31
L'hospital's rule
for type 0/0 or infinity/infinity if f(x) and g(x) differentiable and g'(x) does not equal 0 then lim as x approaches a f(x)/g(x) =lim as x approaches a f'(x)/g'(x) *valid for one-sided limits too*
32
product rule
u'v+v'u
33
quotient rule
u'v-v'u/v^2
34
normal line
perpendicular to tangent line
35
d/dx b^x
b^xlnb
36
d/dxlogbx
1/xlnb
37
chain rule
dy/dx=dy/du du/dx where y=f(u) and u=g(x)
38
chain rule for f(g(h(x)))
apply chain rule twice
39
implicit differentiation
differentiate wrt x then solve for y' (pretend y is an x then add y')
40
logarithmic differentiation
1. take logs of both sides 2. differentiate implicitly wrt x 3. solve for y'
41
general form of taylor series
f(x)=f(a)+f'(a)/1!(x-a)+f''(a)/2!(x-a)^2+...
42
Maclaurin series
taylor series with a=0 sof(x)=f(0)+f'(0)x/1! + f''(0)x^2/2!+...
43
taylor series for f(x)g(x) or f(x)/g(x)
x or dividing the series of f(x) and g(x) usually easier than calculating directly
44
one-to-one
never takes the same value twice horizontal line test
45
f^-1(y)=x implies
f(x)=y
46
graph of inverse function
reflects the graph of f(x) about the line y=x
47
steps for finding inverse function
1. write y=f(x) 2. solve for x in terms of y 3. swap x and y
48
inverse for trig functions
need to restrict domain since trig functions are not one-to-one
49
(f^-1)'(a)=
1/f'(f^-1(a))
50
sinhx=
(e^x-e^-x)/2
51
coshx=
(e^x+e^-x)/2
52
tanhx=
sinhx/coshx
53
sechx
1/coshx
54
cosechx
1/sinhx
55
cothx
1/tanhx
56
point (coshb,sinhb) always lies b=1
right branch of hyperbola x^2-y^2=1 x^2-y^2=cosh^2b-sinh^2b=1
57
sinh^-1x
ln(x+root(x^2+1))
58
cosh^-1x
ln(x+root(x^2-1))
59
tanh^-1x
1/2 ln(1+x/1-x)
60
critical numbers
f'(f)=0 or does not exist
61
rolle's theorem
continuous [a.b] differentiable (a,b) f(a)=f(b) then there exists a c in (a,b) such that f'(c)=0
62
mean value theorem
continuous [a,b] differentiable (a,b) then there exists a c in (a,b) such that f'(c)=f(b)-f(a)/b-a
63
increasing/decreasing test
f'(x)>0 increasing f'(x)<0 decreasing
64
first derivative test
f' +ve to -ve at critical number then f has a local max at c f' -ve to +ve at c, local min
65
concavity
if graph lies above all tangent, concave upward (smiley) below, concave downward :(((
66
conditions for concave up/down
f''(x) > 0 for all x, concave upward :) f''(x)<0 for all x, concave downward :(
67
second derivative test
if f'(c)=0 and f''(c)>0, local min at c if f'(c)=0 and f''(x)<0, f has local max at c
68
position vector
origin to initial point
69
|a|
root a1^2+a2^2+...
70
unit vector
u=a/|a|
71
basis vectors
i,j,k <1,0,0> <0,1,0> <0,0,1>
72
dot product
a.b=a1b1+a2b2+a3b3 a.b=|a||b|cos theta
73
properties of dot product
a.a = |a|^2 a.b=b.a (ca).b=c(a.b)=a.(cb) where c is a constant
74
vectors orthogonal if
theta=pi/2 therefore a.b=0
75
alpha,beta and gamma are
angles a vector makes with the positive x,y, and z axes respectively. cos alpha= a1/|a| cos beta = a2/|a| cos gamma = a3/|a| where a =
76
compa b
a.b/|a|
77
proja b
(a.b/|a|)a/|a|=a.b/|a|^2 a
78
cross product
result is vecotr axb is 3x3 matrix determinant axb=|a||b|sin theta
79
properties of cross product
axb orthogonal to both a and b axb=-bxa (ca)xb=c(axb)=ax(cb) where c is a constant ax(b+c)=axb+axc (a+b)xc=axc+bxc
80
vectors parallel if
axb=0
81
length of axb ie |axb|
area parallelogram
82
a.(bxc)=
(axb).c
83
Vparallelepiped
|a.(bxc)|
84
vector triple product ax(bxc)=
(a.c)b-(a.b)c
85
equation of line
r=ro+tv where ro is position vector of Po v is parallel to L
86
parametric equation of line
x=x0+at y=y0+bt z=z0+ct
87
symmetric equations of line
x-x0/a=y-y0/b=z-z0/c
88
lines are parallel if
direction vectors parallel
89
lines intersect if
components of vector equation of each line match at P
90
skew lines
do not intersect and are not parallel
91
vector equation of plane
n.(r-r0)=0 n.r=n.ro ax+by+cz=d where d=ax0+by0+cz0
92
two planes parallel if
normal vectors are parallel
93
if two planes are not parallel
they intersect in a straight line
94
to find angle between planes
1. find normal vectors 2. cos theta = n1.n2/|n1||n2| (or use sin and the cross product)

Maths 1 (5 decks)

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